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Linear Algebra
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Fatemeh Hamidi Sepehr, Erchin Serpedin
Often it is necessary to extend the concept of matrix invertibility to a wider class of rank-deficient matrices by means of concepts such as generalized inverse or pseudoinverse of a matrix. One of the major applications of the concept of generalized inverse is in solving systems of equations of the form: Ax = b, where matrix A is neither square nor nonsingular. Such a system of equations may not admit any solution or may admit an infinite number of solutions.
Priority-based state machine synthesis that relaxes behavior design of multi-arm manipulators in dynamic environments
Published in Advanced Robotics, 2023
Let us assume that , in order to consider a redundant robot, which has large DoF. If is full row rank, namely , we can solve the linear Equation (2) with the generalized inverse matrix as The generalized inverse matrix is defined as a matrix satisfying the following condition [13]: Note that is not unique because the matrix has the null space. Thus, Equation (3) can represent all solutions of the inverse problem of (2). Although the following discussion does not depend on the choice of , we select the pseudo-inverse , one of the generalized inverse matrices, to use for simplicity. In implementation, we can use regularization methods to stabilize numerical calculation of the pseudo-inverse matrices (see discussion in [14] for more detail).
Generalized formulation of extended cross-section adjustment method based on minimum variance unbiased linear estimation
Published in Journal of Nuclear Science and Technology, 2019
Kenji Yokoyama, Takanori Kitada
To express the solutions for Equation (107), we introduce a generalized inverse for a singular matrix, which includes a rectangular matrix. Note that the generalized inverse of a matrix is not unique. The Moore–Penrose pseudoinverse is a special case of the generalized inverse, and is determined uniquely. By definition, for the generalized inverse of an arbitrary matrix , we have